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All automorphisms of the Calkin algebra are inner. (English) Zbl 1250.03094
Summary: We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a construction by N. C. Phillips and N. Weaver [Duke Math. J. 139, No. 1, 185–202 (2007; Zbl 1220.46040)] of an outer automorphism using the continuum hypothesis, this gives a complete solution to a problem of L. G. Brown, R. G. Douglas and P. A. Fillmore [Ann. Math. (2) 105, 265–324 (1977; Zbl 0376.46036)]. We also give a simpler and self-contained proof of the Phillips-Weaver result.

MSC:
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
46L05 General theory of \(C^*\)-algebras
46L40 Automorphisms of selfadjoint operator algebras
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